Constructing continuum many countable, primitive, unbalanced digraphs

Crown-free highly arc-transitive digraphs

“…The following is as in [5,Lemma 2.4]. Since A; B v C and C 2 C m , the intersection desc.a i ; b j / is finitely generated for all i; j .…”

Infinite primitive and distance transitive directed graphs of finite out-valency

“…During this process, there will be countably many 'tasks' to be performed: there are countably many choices of f.g. A in each D i and countably many isomorphism types of ≤ + -embeddings f : A → B with B ∈ C Γ (by Lemma 3.1). As we have countably many steps available during the construction, it will suffice to show how to complete one of these tasks: ensuring that they are all completed during some stage of the construction is then just a matter of organisation (see the proof of Theorem 2.8 of [7] for a formal way of doing this).…”

“…In these examples, the descendant sets are directed trees, and the resulting examples are also highly arc transitive. Similar methods were used in [7] to construct continuum-many non-isomorphic countable, primitive, highly arc transitive digraphs all with isomorphic descendant sets. So this suggests that a classification of such digraphs is out of the question, even under the very strong assumption of high arc transitivity.…”